Hi, I am a Japanese and a "sake" lover.
Your understanding of the meaning and the difference between "sake" and "nihonsyu" is right.
The word "sake" had long been used in Japan in the meaning of "rice wine" naturally, because there had not been other alcoholic drinks in history, but along with the generalization of the habit, since the time of the introduction of western life style in the Meiji era, especially after the WWII and especially in recent decades, to take to drinking other alcoholic drinks of foreign origin such as beer, (grape)wine, whisky etc., we gradually regard "rice wine" simply as one of the alcoholic drinks, though keeping in mind, unconcsiously maybe, "sake" is is THE Japanese traditional and sacred alcoholic drink. So, different reactions you experienced reflect this unconscious mind-set of Japanese people.
Personally, I drink sake (rice wine),(grape)wine, beer, shochu, sometimes Chinese rice wine... When I say to my dear wife: "Today, I'll drink sake", it means for me automatically "rice wine" and my wife (we are an aged couple) asks me to confirm what "sake", id. "rice wine" or "grape wine" or "beer". After my answer, she will prepare fortunately for me, depending what I choose, food which suits the drink (what a lucky man I am!).
So on my first trip to Japan, when I wanted "rice wine" (as defined above), I'd say: "Sake o kudasai" (or somesuch). But the Japanese server would always reply: "Biiru? Whisky?". Then I'd be confused - I've asked for rice wine, why is he asking if I want beer or whisky?
You don't look like a Japanese physically at a first glance, do you?
I don't know this happens even now, but before a while, the attitude of the Japanese server you mentioned might be natural and frequent. Because Japanese people didn't know that foreign people were interested in sake (rice sake) and moreover in Japanese cooking. In reality, this was true, I think. This fashion, or interest of foreign people for sake is very recent. It's one of the effects of Japanese cuisine popularity in foreign countries, such as sushi, sashimi, and moreover Japanese style of cooking in general. So, the server never thought you would take other drinks than beer or whisky which were nothing but alcoholic drinks of foreign origine.
In this context I have explained, the word "nihonshu" has a reason to be used, for "nihonshu" means literally "Nihon" (Japan) and "shu" (on-yomi of the kanji ๐Aalcoholic drink in general), id. Japanese alcoholic drink. Certainly, this way of calling our own traditional sake "Japanese alcoholic drink" in comparison with other foreign drinks is to be understood, but not pleasant pychologically for us, at least for old generations, or for certain masters of old fashioned "Izakaya". In brief, the word "nihonshu" is forced to be used, at least for me, to make clear what you want is not alcoholic drinks in general (or such as beer, grape wine, whisky etc.), but our "Japanese fermented rice alcoholic drink".
However, occasionally, when I asked for "nihonshu", the server would have absolutely no idea what I meant. But if I changed to asking for "sake", he'd immediately say: "Ah! Sake! Hai!" - then give me rice wine! I also saw a Japanese TV show, where a Japanese actor, talking to other Japanese actors, asked for "sake" - not "nihonshu" - and was immediately given rice wine.
It's a little hard to imagine the situations, but these two situations seem to have a different connotation.
The former is unlikely, but I imagine he might be one of these Japanese who had the mind-set I have explained. I don't know. Or he might simply be embarassed to hear what was unexpected.
The latter depends on situations or locations where this conversation were exchaged. If the location was where Japanese food was served, "sake" might mean naturally and automatically "nihonsyu", because all the people of the show had the common understanding what was to be served there.
Anyway, toast to sake lovers!